An intelligent optimization method for highway route selection based on comprehensive weight and TOPSIS

In order to accurately analyze and evaluate multi-index and multi-route traffic schemes for comparison and selection, we introduce herein a comprehensive weight and an intelligent selection algorithm for traffic scheme optimization to improve upon the shortcomings of common qualitative and quantitative analysis methods. Firstly, we establish an evaluation index system of transportation by traffic scheme considering the factors of technology, ecological environment, social environment, and economy, based on the whole life cycle. Secondly, the comprehensive weight based on subjective and objective factors is constructed. Finally, we establish an optimization method for transportation schemes by using the comprehensive weight and Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) model. The results show that the evaluation index system based on the whole life cycle is more comprehensive and accurate. The comprehensive weight vector avoids the defects of single weight methods and makes full use of subjective data and expert opinions. The comprehensive weight vector is introduced into the decision-maker’s preference coefficient, so that analysts can determine the scheme according to the subjective and objective information and to the required accuracy. This method uses a large number of evaluation groups to evaluate the scheme, and the evaluation results show greater objectivity and efficiency.


Introduction
The selection of highway route schemes directly influences construction costs; construction and operation safety; the length of the construction period; and the costs relating to operation, management, and maintenance. In practical work, a trial-and-error method based on the work experience of designers and on expert evaluation is not time-consuming, but it can easily miss optimal schemes, and this process cannot be evaluated by experts and stakeholders with a large amount of data. For transportation schemes with complex geographical and societal considerations, it is more difficult to adapt this method to the requirements of scheme optimization. Therefore, a new intelligent optimization method is needed to quickly and comprehensively process data from a large number of evaluators and select the optimal scheme. system generally uses the economic, technical, and environmental indicators during the construction period [11,[23][24][25][26][27][28]. It does not consider safety factors, and it lacks indicators of maintenance and management costs during the operation period; this cannot accurately and comprehensively reflect the scheme. In order to comprehensively consider the indicators of the scheme, we used the AHP to analyze factors influencing the scheme and built a full-lifecycle index system for highways that considers technology, the ecological environment, the social environment, and engineering economy (see Table 1). Indexes can be divided into qualitative indexes and quantitative indexes based on their classification methods. According to trend, they can be divided into positive and negative categories; for a positive index (marked with "+"), a scheme will be more valuable when it is larger, and for a negative index (marked with "-"), a scheme will be more valuable when it is smaller.
Quantitative indexes. Quantitative indexes are analyzed according to their corresponding values. If there are m evaluation schemes and n quantitative indexes, and each scheme has a quantitative value B ij for each quantitative index (e.g., scheme length or construction costs), then For an index with a negative tendency, positive processing (similar to a tendency) [29] is required: As an example, take three design schemes that were formulated for a certain highway with construction costs (CNY 100 million) of Qualitative indexes. Qualitative indexes are obtained from fuzzy classifications in the data. The fuzzy classifications have nine levels each: excellent, good, medium, bad, and poor, and large, general, and small. The evaluation scale used is the (1/9, 9) scale method.
Weight vectors of the index system. By getting the score of each index according to the rating method of the evaluators and normalizing the weight of each index [30], subjective randomness and preference can be inferred to a certain extent; this is called the subjective weight. Objective weight is calculated from quantitative indexes. To consider an evaluator's subjective perception of indexes and objective information among indexes, as well as an analyzer's preferences, this paper introduces a new method for determining weights, i.e., a comprehensive weight method.
Calculating the subjective weight of an index. The subjective weight is calculated via the analytic hierarchy process (AHP) [31], which is scored by the evaluators according to laws, regulations, experience, and interests. The steps are as follows.
Step 1: Create the weight judgment matrix.
A hierarchy model is established to compare the indexes of the criterion layer, and the relative importance of the indexes is obtained. The judgment matrix is constructed by the (1/9, 9) scale method. See Table 2 for the general form of the judgment matrix.
Step 2: Calculate the relative weight of each factor at each level.
According to the judgment matrix, the eigenvalues and the maximum eigenvalues of each factor are calculated by the square root method.
Here, M i is the row element product of the row in which the i th index of the k th evaluator in the judgment matrix is located; Z k i is the relative weight of Z k i after normalization; η k is the eigenvector; l k max is the maximum eigenvalue.
Step 3: Calculate the consistency index.
To ensure that a weight is reasonable, it is necessary to check the consistency of each judgment matrix to determine whether it has satisfactory consistency. If it does not, the judgment matrix should be modified until it meets the consistency requirements. The formula for checking consistency is as follows: where K is the consistency index; G is the random consistency ratio; E is the order of the judgment matrix; R is the random consistency index corresponding to the order of the judgment matrix.
Step 4: Find the subjective weight vector.
From the weight and the average social impact weight of each evaluation, the subjective weight can be calculated as follows: Calculating the objective weight of an index. Information entropy is used for finding the objective weight. The data in this method are obtained from a numerical analysis of the highway route scheme evaluation index system.
Step 1: Generate the decision matrix. If there are m evaluation schemes and n evaluation indexes, then d ij (i = 1, 2, . . .,m; j = 1, 2, . . .,n), and the index weight matrix D can be In Table 2, h is the total number of evaluators; k denotes the k th evaluator; n is the total number of indicators in the criteria layer; F i k and F j k are the i th and j th indicators of the k th evaluator in the criteria layer, respectively; and f ij k is the importance of indicator F i k compared with indicator F j k . The value is determined according to the (1/9, 9) scale method by each evaluator.
https://doi.org/10.1371/journal.pone.0262588.t002 expressed as Step 2: Normalize the decision matrix. To eliminate the influences of each evaluation index on the evaluation of the route scheme due to the different dimensions of each evaluation index, normalization is required. In normalization, a decision matrix D generates a stan- The normalized value is found as follows: Step 3: Calculate the weight. If the feature weight of the i th evaluation object is P ij under the j th index, then Step 4: Calculate the entropy e j of the j th index: Step 5: Calculate the coefficient of difference d j for the j th index. For a certain index d j , the smaller the difference v ij is, the larger d j will be. When the values of the j th indexes for each evaluated object are equal, then e j = e max and d j will be Step 6: Calculate the entropy weight of each index: where P n j¼1 m j ¼ 1; m j �0 (j = 1, 2, . . .,n).

Comprehensive weight.
To consider an evaluator's subjective perception of indexes and objective information among indexes, as well as an analyzer's preferences, we introduce a new method for determining weights, i.e., a comprehensive weight method.
If the comprehensive weight of each index is expressed as where P m j¼1 o l ¼ 1, ω j � 0 (j = 1, 2, . . ., m), z ij is the evaluation matrix after standardization, then the comprehensive weighted score of a scheme is To both give consideration to subjective preference (for a subjective or objective weighting method) and make full use of the information provided by the subjective weighting method and the objective weighting method, and thereby achieve unity of the subjective and objective methods, the following optimization decision-making model is established: where 0 � ρ � 1 reflects the decision-maker's preference for subjective weight, named preference coefficient. The higher the value of ρ is the decision maker prefer subjective weight. Theorem 1 (S1 Appendix): If P m i¼1 z 2 ij > 0 (j = 1, 2, . . ., n), then the optimization formula (15) has a unique solution, which is TOPSIS evaluation model. Many methods can be used in optimal selection, such as TOPSIS, LBWA [32], FUCOM [33] or BWM models. However, considering that TOPSIS is simpler and more suitable for internal sorting of sample data, this paper adopts TOPSIS. TOP-SIS [34,35] refers to the technique for order preference by similarity to an ideal solution, the basic concept of which is to determine the optimal solution and the worst solution for a normalized original data matrix, then calculate the distance between the evaluated solution and the optimal solution and the worst solution, obtain the nearness degree between the evaluated solution and the optimal solution, and, on this basis, assess the advantages and disadvantages of each evaluated object. This method is widely used in multi-objective scheme selection, such as for highways, electric power, etc. [17,[36][37][38][39].
Step 1: If there are n evaluation objects and m evaluation indexes, then we can obtain an m×n initial judgment matrix V: Step 2: The dimension of each index may be different, so that decision matrix should be normalized: . . Where Step 3: Calculate the weight vector of each index from Eq (16) and generate a weighted judgment matrix.
Step 4: Calculate the positive and negative ideal solutions of the evaluated targets according to the weighted judgment matrix. The positive ideal solution is z þ j : ( The negative ideal solution is z À j : ( where j + refers to the benefit index; j − refers to the cost index.
Step 5: Calculate the Euclidean distance between each target value and the ideal value, S þ i ; S À i .
S À i ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi X m Step 6: Calculate the relative degree of closeness of each target, C i Step 7: Sort the targets based on the relative degree of closeness and generate the decision criteria. When the C i + value approaches 1, the evaluation object becomes closer to the positive ideal solution.

Highway route scheme optimization algorithm
First, the indicator system of the highway route scheme can be constructed according to the characteristics of the project. All or some of the indicators in Table 1 can be selected, and some indicators can be added according to the characteristics of the project. The qualitative indexes and quantitative indexes in the index system are processed to obtain quantitative and positive index data and to build the initial judgment matrix. Second, the subjective weight and objective weight are calculated. Thirdly, the decision maker decided the preference coefficient and calculates the comprehensive weight.
Finally, the TOPSIS model is used to calculate the relative closeness of each scheme, and the optimal scheme is obtained (Fig 1).

Route scheme selection automation
Highway route optimization algorithm involves a lot of calculation, especially when the number of experts and stakeholders is large. Therefore, it is necessary to use computer to calculate quickly to obtain the optimal scheme (Fig 2).

Discussion and empirical research
The intelligent optimization method proposed in this paper establishes an index system based on the whole life cycle, which is comprehensive and avoids the adverse impact of incomplete index system on the results. This method takes into account both subjective weight and objective weight, and the two weights are calculated independently. At the same time, in order to consider the focus of decision-makers, preference coefficient is introduced. This method uses computer technology, can adapt to a large number of evaluation population, and can quickly obtain the best scheme after evaluation. It can not only be used in highway scheme selection, but also can be used in other multi-objective scheme selection.
We took the route scheme for a highway in Tibet as an example for evaluation and comparison and obtained the optimal scheme through calculation. The length of the corridor from the Lalangqu to Zhagongqu is approximately 12 km, and it ascends the Xuegula mountain. Due to the complicated topography and geology of the area, a large average longitudinal slope, and a high proportion of bridges and tunnels, three schemes were formulated for comparison. Fig 3 shows the comparison and selection of highway route schemes.
Step 1: We obtained the index system data (the qualitative indexes and the quantitative indexes) of a multi-index highway route scheme, and then we built a judgment matrix. See Table 3 for the indexes.
Step 2: Twenty-three evaluators, comprising 7 experts and 16 stakeholders, scored the weight of each evaluation index. We calculated the comprehensive weight according to formulas (5), (12), and (16), as shown in Table 4. According to the decision-maker's preference for qualitative and quantitative indexes, the preference coefficient ρ was 0.6.
Step 3: We normalized the judgment matrix and used the calculated comprehensive weight vector to build a weighted decision matrix. See Table 5 for the results.
Step 4: We calculated the relative degree of closeness between each scheme and the ideal solution. See Table 6 for the results.
In order to analyze the sensitivity of preference coefficient, we calculated the closeness under different preference coefficients, as shown in Fig 4. It can be seen from Fig 4 that each scheme has a different closeness with different preference coefficient, and the optimal scheme is finally selected. S1 with high closeness shall be preferred, when ρ 2 (0,0.651). S2 with high closeness shall be preferred, when ρ2 (0.0651,1). It shows that the preference coefficient can affect and determine the conclusion of multi-objective scheme selection, that is an important sensitive factor in multi-objective scheme selection. The CS1 inversly decreased gradually along with the increase of ρ. It suggests that the main reason was that S1 has small scale, low cost and obvious advantages in overall objective indicators, but it covers a lot of land and has a long route, which leads to the low score of the evaluators. Along with ρ increase, C S2 and C S3 are increasing trends, but the increase of S2 is greater, indicating that S2 is right ρ More sensitive. The analysis shows that the main reason is that although S2 tunnel is long, the route mileage is short, avoiding basic farmland, the cost and technical indicators are higher than S3, resulting in higher scores for the evaluators.
There are fewer human factors involved in the objective weight, and more evaluators in the objective weight. Considering from the perspective of economy, technology and stakeholders, the two weight schemes have their own advantages and disadvantages. Decision makers have a deeper understanding of the project and give preference factors based on the overall judgment of subjective and objective factors, considering their own interests, the environment during the construction period and the quality of evaluators. Decision makers can not easily judge the impact of preference factors on project conclusions, which not only considers the subjective and objective indicators and the decision-makers' evaluation of subjective and objective indicators, but also avoids the decision-makers from giving intuitive conclusions, which is conducive to ensuring the scientificity of scheme selection. If construction cost C41, operation, management, and maintenance costs C42 and vehicle operation costs C43 are ignored, then the relative degrees of closeness are in the order C S3 > C S2 > C S1 (Table 7). Therefore, the construction of the indexes directly influences the evaluation results.
Through the above analysis and calculation, S3 is the optimal scheme. At the same time, the expert scoring method and the general comprehensive weight method are compared. This method is more in line with the expectations of the construction unit and experts.

Conclusions
In this paper, we discussed the limitations of highway route selection. To address these limitations, we built an evaluation index system for highway route schemes based on the full life cycle of the route and proposed a new weight calculation method that uses the preference of decision-makers and comprehensively calculates the subjective weight and objective weight. We used a comprehensive weight vector and the TOPSIS model to build an evaluation system for a highway route scheme, and we verified the validity of this method of scheme evaluation using examples. The following conclusions can be drawn: The construction of evaluation indexes directly influences the results of scheme evaluation. When subsequent operations, management, and maintenance indexes are included based on the full life cycle, the evaluation results are more accurate.
The comprehensive weight vector can overcome the defects of the single weight method; it not only makes use of objective data, but also fully considers experts' opinions. The comprehensive weight vector increases the preference coefficient ρ, which is essential different from existing other comprehensive weight methods. The preference coefficient ρ can be determined by the accuracy of subjective and objective information and preference. When the decisionmaker obtains the subjective weight and objective weight, through comprehensive analysis of the project, a preference coefficient from the perspective of the interests of the decision-maker is proposed, which can better serve the decision-maker.
The evaluation results for the selected route scheme based on the comprehensive weight model with TOPSIS were basically consistent with on-site conditions; therefore, the method is feasible and valid for comprehensively evaluating a route scheme.  Table 6. Evaluation results of the relative degree of closeness.

S1 S2 S3
Relative degree of closeness 0.853 0.841 0.849 The degrees of closeness are in the order C S1 > C S3 > C S2 , so scheme S1 is the best, followed by scheme S3 and scheme S2. The evaluation results are completely consistent with on-site conditions.
https://doi.org/10.1371/journal.pone.0262588.t006 Intelligent computing can be used to form an evaluation of the scheme by a large number of evaluation groups (including experts, stakeholders, etc.), and the evaluation results are more objective. Intelligent Optimization Method can not only be used in highway scheme selection, but also can be used in other multi-objective scheme selection.
In future work, we will further study intelligent optimization methods of highway schemes. For example, we will study the method of selecting project indicators, differentiate the weight calculation methods of experts and stakeholders, and explore whether there is a better evaluation model that could replace TOPSIS.In addition, we will assess the usability of the approach for domain experts. Our ultimate objective is to use our scheme optimization approach for automated incident reporting, which can be made intelligent. The system only needs to input the index of the scheme and the score of the evaluation group, and the preference coefficient of the decision-maker can be used to automatically obtain the optimal scheme. To achieve this aim, we will use examples of potential selected schemes to evaluate the applicability of this method and to identify monitoring activities that may be useful in detecting or investigating the selection of these schemes.